SIMILAR, SIMILAR FIGURES (Geometry). Similarity. resem blance, or likeness, means sameness in some, if not in all, particulars.
In geometry, the word refers to a sameness of one particular kind.
The two most important notions which the view of a figure will give are those of use and *Ave, ideas which have no oonnection whatsoever with each other. Figures of different sizes may have the same shape, and figures of different shapes may have the same size. In the latter case they are called by Euclid equal, in the former similar (similar figures, koala ex-boa-re). The first term [EQUAL; RELATION], In Euclid's first use of It, includes united sameness both of size and shape; but he soon drops the former notion, and, reserving equal to signify sameness of size only, introduces the word similar to denote sameness of form : SO that the equality of the fundamental definition is the subsequent combined equality and similarity of the sixth book.
Similarity of form, or, as we shall now technically say, similarity, is a conception which is better defined by things than by words; being in fact one of our fundamental ideas of figure. A drawing, a map, a model. severally appeal to a known idea of similarity, derived from, it may be, or at least nourished by, the constant occurrence in nature and art of objects which have a general, though not a perfectly mathe matical, similarity. The rudest nations understand a picture or a map almost instantly. It Is not necessary to do more in the way of defini tion, and we must proceed to point out the mathematical tests of similarity. We may observe indeed that errors or monstrosities of size are always more bearable than those of form, so much more do our conceptions of objects depend upon form than upon size. A painter may be obliged to diminish the size of the minor parts of his picture a little, to give room for the more important objects : lint no one ever thought of making a change of form, however slight, in one object, for the sake of its effect on any other. Tho giant of Rabelais, with whole nations carrying on the business of life inside his mouth, is not so monstrous as it would have been to take the ground on which a nation might dwell, England, France, or Spain, invest it with the intellect and habits of a human being, and make it move, talk, and reason : the more tasteful fiction of Swift is not only bearable and conceivable, ,e but has actually made many a simple person think it was meant to be taken as a true history.
Granting then a perfect notion of similarity, we now ask in what way it is to be ascertained whether two figures are similar or not. To simplify the question, let them be plane figures, say two maps of England of different size., but made on the same projection. It is obvious, in the first place, that the lines of one figure must not only be related to one another in length in the same manner as in the other, but also in position. Let us drop for the present all the curved lines of the coast, &c., and consider only the dots which represent the towns. Join every such pair of dots by straight lines : then it is plain that similarity of form requires that any two lines in the first should not only be in the samo proportion, as to length, with the two corre sponding lines in the second, but that the first pair should incline at the same angle to each other as the second. Thus, if L Y be the line which joins London and York, and r o that which joins Falmouth and Cheater, it is requisite that La' should be to rein the game proportion in the one map that it is in the other ; and if F c produced meet L Y produced in o, the angle o o 1r in one map mud be the same as in the other. Hence, if there should be 100 towns, which are therefore joined two and two by 4950 straight lines, giving about 12 millions and a quarter of pairs of lines, it is clear that we must have the means of verifying 12j millions of proportions, and as many angular agree nienta. Bit if it be only assumed that similarity is a possible thing, it is easily shown that this large number is reducible to twice 9a. Let it be granted that 1 y on the windier map is to represent L Y on the larger. Lay down f and c in their proper places on the smaller reap, each with reference to I and y, by comparison with tho larger map : then f and c are in their proper places with reference to each other. For if not, one of them at least must be altered, which would disturb the correctness of it with respect to f and y. Either then there is no such thing an perfect similarity, or else it may lc entirely ob tained by comparison with 1 and y only.